Two metal rods, one silver and the other copper, are both attached to a steam chamber, with a temperature of 100°C, at one end, and an ice water bath, with a temperature of 0°C, at the other. The rods are 5.0 cm long and have a square cross-section, 2.0 cm on a side. How much heat flows through the two rods in 60 s? The thermal conductivity of silver is 417 W/(m·K), and that of copper is 395 W/(m·K). No heat is exchanged between the rods and the surroundings, except at the ends.

Question

Quizplus · Accepted Answer

First, we need to find the rate of heat transfer through each rod using Fourier's Law:

$q_A=\frac{k_A A \Delta T}{L}$
$q_C=\frac{k_C A \Delta T}{L}$

where $q$ is the rate of heat transfer, $k$ is the thermal conductivity, $A$ is the cross-sectional area, $\Delta T$ is the temperature difference, and $L$ is the length of the rod.

For both rods, $\Delta T=100°C$ (the temperature difference between the steam chamber and the ice water bath). Substituting the given values:

$q_A=\frac{(417	ext{ W/(m·K)}) (0.02	ext{ m})^2 (100	ext{ K})}{0.05	ext{ m}}=66.72	ext{ W}$
$q_C=\frac{(395	ext{ W/(m·K)}) (0.02	ext{ m})^2 (100	ext{ K})}{0.05	ext{ m}}=63.36	ext{ W}$

The total rate of heat transfer through both rods is $q=q_A+q_C=130.08	ext{ W}$. Over 60 s, the total heat transfer is:

$Q=q \Delta t=130.08	ext{ W} 	imes 60	ext{ s}=7804.8	ext{ J}=7.8	ext{ kJ}$

This is closest to choice B, 39 kJ, which is the correct answer.

Two Metal Rods, One Silver and the Other Copper, Are